The number of staircase walks on a grid with m horizontal lines and n vertical lines is given by (m + n m) = ((m + n)!)/(m!n!) (Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch 2003). The first few values for m = n = 1, 2, ..., are 1, 2, 6, 20, 70, 252, ... (OEIS A000984), which are the central binomial coefficients. A Dyck path is a staircase walk from (0, 0) to (n, n) which never crosses (but may touch) the diagonal y = x.

Catalan number | central binomial coefficient | Dyck path | Ferrers diagram | staircase polygon